Hausdorff andHausdorff andNon-Hausdorff Non-Hausdorff

SpacesSpaces

Mayra IbarraMayra Ibarra

MATH101MATH101

What is a Topological What is a Topological Space?Space?

Recall:Recall: A topological space is an ordered pair (X, Ƭ), A topological space is an ordered pair (X, Ƭ),

where X is a set and T is a collection of subsets of where X is a set and T is a collection of subsets of X such thatX such that

i.i. Ø Ø ЄЄ Ƭ Ƭ and X and X ЄЄ Ƭ Ƭ

ii.ii. UU∩V ∩V ЄЄ Ƭ Ƭ whenever U, V whenever U, V ЄЄ Ƭ Ƭ;;

iii.iii. ∪∪{ U{ Ua a : a : a ЄЄ II} } ЄЄ Ƭ whenever {U Ƭ whenever {Ua a : a : a ЄЄ II} } Ƭ.⊂ Ƭ.⊂ 11

Separation AxiomsSeparation Axioms22

T0 : If a, bϵX, there exists an open set OϵƬ such that either aϵO and b O, or bϵO and a O.

T1 : If a, bϵX, there exist open sets Oa, ObϵƬ containing a and b respectively, such that b Oa and a Ob.

T2 : If a, bϵX, there exist disjoint open sets Oa and Ob

containing a and b respectively.

T3 : If A is a closed set and b is a point not in A, there exist disjoint open sets OA and

Ob containing A and b respectively.

X

T4 : If A and B are disjoint closed sets in X, there exist disjoint open sets OA and OB containing A

and B respectively

X

T5 : If A and B are separated sets in X, there exist disjoint open sets OA and OB containing A and B respectively

Hausdorff SpacesHausdorff Spaces All metric spaces are HausdorffAll metric spaces are Hausdorff33

The real number line with the usual topology is The real number line with the usual topology is HausdorffHausdorff44

Suppose we have an open set, S, of real numbersSuppose we have an open set, S, of real numbers For each For each ppЄЄS we can find an S we can find an εε>0, such that the >0, such that the εε--

neighborhood of neighborhood of pp forms an open set inside of S. forms an open set inside of S. Suppose Suppose p, qp, q ЄЄ S and are separated by a distance r. S and are separated by a distance r.

We can pick We can pick εε1 1 and and εε22 for for p p andand q q, respectively, such , respectively, such

that r/2 ≥ that r/2 ≥ εε11 > 0 and r/2 ≥ > 0 and r/2 ≥ εε22 > 0. Thus > 0. Thus p p and and qq will be will be

separated by disjoint open sets. separated by disjoint open sets.

Subspaces of Hausdorff Subspaces of Hausdorff Spaces are also HausdorffSpaces are also Hausdorff55

If Y If Y X and (X, d) is Hausdorff, then ⊂ X and (X, d) is Hausdorff, then ⊂any two distinct points xany two distinct points x11, x, x2 2 ЄЄ Y will Y will also be in X. Since xalso be in X. Since x11and xand x22 are in X, are in X, there are disjoint open sets Othere are disjoint open sets O11 and and OO2 2 containing xcontaining x11and xand x22, respectively. , respectively. Thus Y contains points that can be Thus Y contains points that can be separated by disjoint open sets and separated by disjoint open sets and is Hausdorff.is Hausdorff.

Non-Hausdorff SpacesNon-Hausdorff Spaces Zariski TopologyZariski Topology Let X be [0, 1] [2, 3]. Define the following ∪Let X be [0, 1] [2, 3]. Define the following ∪

equivalence relation on X:equivalence relation on X: a ~ a + 2a ~ a + 2 for all 0 ≤ a ≤ 1 except for a = ½for all 0 ≤ a ≤ 1 except for a = ½ b ~ b – 2 b ~ b – 2 for 2 ≤ b ≤ 3 except for b = 5/2for 2 ≤ b ≤ 3 except for b = 5/2

[0] [1]

5/2

1/2

Proof:Proof:Notice that every point in [0, 1] is Notice that every point in [0, 1] is

identified with a point of [2, 3] except fpr the identified with a point of [2, 3] except fpr the two points ½ and 5/2, which remain two points ½ and 5/2, which remain unidentified. Thus are contained in X/~. A unidentified. Thus are contained in X/~. A set in X/~ is open iff it is open in X. set in X/~ is open iff it is open in X. However, we see that there are no disjoint However, we see that there are no disjoint open sets that contain ½ and 5/2. So X is open sets that contain ½ and 5/2. So X is Non-Hausdorff when considered under this Non-Hausdorff when considered under this equivalence relation.6equivalence relation.6

Regular and Normal Regular and Normal SpacesSpaces77

Regular SpaceRegular Space8-8- A space which is A space which is both a Tboth a T11 and a T and a T33 space space

Normal SpaceNormal Space99-A space which is both -A space which is both a Ta T11 and a T and a T44 space space

Every Normal Space is RegularEvery Normal Space is Regular Every Regular Space is HausdorffEvery Regular Space is Hausdorff Not all Hausdorff Spaces are RegularNot all Hausdorff Spaces are Regular In the definitions of Normal and Regular we In the definitions of Normal and Regular we

can replace the use of disjoint open sets, can replace the use of disjoint open sets, and use disjoint closures to yield the same and use disjoint closures to yield the same spacesspaces

The same is not true for the definition of The same is not true for the definition of HausdorffHausdorff

Completely HausdorffCompletely Hausdorff1010

TT2 ½2 ½ Axiom- If a and b are two points of a Axiom- If a and b are two points of a

topological space X, there exist open topological space X, there exist open sets Osets Oa a and Oand Ob b containing a and b, containing a and b,

respectively, such that Ōrespectively, such that Ōaa ∩∩ Ō Ōb b = Ø= Ø

A space that satisfies this axiom is called A space that satisfies this axiom is called a completely Hausdorff space.a completely Hausdorff space.

Normalregular

Completely Hausdorff

Hausdorff

Compact Hausdorff Compact Hausdorff SpacesSpaces

If a Hausdorff space is compact, If a Hausdorff space is compact, then it is normal, and therefore then it is normal, and therefore also regular.also regular.

Let the sequence (SLet the sequence (Snn) in a Hausdorff ) in a Hausdorff

Space have a limit, s. Suppose that the Space have a limit, s. Suppose that the sequence has another limit, p, such that sequence has another limit, p, such that p≠s. Let U and V be disjoint p≠s. Let U and V be disjoint neighborhoods of s and p, respectively. neighborhoods of s and p, respectively. Then if we choose a sufficiently large n, Then if we choose a sufficiently large n, SSnnЄЄU however, for a sufficiently large n, U however, for a sufficiently large n,

SSnnЄЄV. This contradicts the Hausdorff V. This contradicts the Hausdorff

property because Uproperty because U∩V ∩V ≠ Ø. Thus the ≠ Ø. Thus the sequence cannot have more than one sequence cannot have more than one limit. limit.

If X is Hausdorff and (SIf X is Hausdorff and (Snn) ) is a sequence in X that is a sequence in X that

converges to a point sconverges to a point sЄЄX, X, and if y is an and if y is an

accumulation point of the accumulation point of the set set {S{Snn | | n = 1, 2, . . .}, then s = y. n = 1, 2, . . .}, then s = y.

Suppose s≠y. Then there exist open sets U, V Suppose s≠y. Then there exist open sets U, V ЄЄ X X for s and y respectively such that U∩V = Ø. Also, for s and y respectively such that U∩V = Ø. Also, since (Ssince (Snn) converges to s, there exists a natural ) converges to s, there exists a natural

number, N such that n>N implies that Snumber, N such that n>N implies that Sn n ЄЄ U. U.

Let Let i i be such that 1< be such that 1< i i < N, and let W< N, and let Wii , V , Vii ЄЄ X be X be

open sets for sopen sets for sii and y, respectively, such that W and y, respectively, such that W ii ∩ V ∩ Vii

= Ø, unless y= s= Ø, unless y= sii for some for some i.i.

Define VDefine Vii’= V’= Vi i in case y≠sin case y≠sii , and V , and Vii’= V in case y=s’= V in case y=sii

and define V’ as the intersection of V with a finite and define V’ as the intersection of V with a finite collection of open sets, Vcollection of open sets, Vii’. So y ’. So y ЄЄ V’ V’ ЄЄ a system of a system of

neighborhoods around y. neighborhoods around y.

Let z be such that zLet z be such that zЄЄV’ and y≠z, then z≠sV’ and y≠z, then z≠si i for any for any

i i since if since if i i > N, s> N, si i ЄЄU and U∩V’ U∩V = Ø, ⊆U and U∩V’ U∩V = Ø, ⊆

And ifAnd if i ≤ i ≤ N, then either y=sN, then either y=sii ≠z, or s ≠z, or siiЄЄWWii , and W , and Wii

∩V’ W⊆∩V’ W⊆ ii ∩ V ∩ Vii = Ø. = Ø.

Thus there is a neighborhood V’ of y, such that Thus there is a neighborhood V’ of y, such that no point z ≠ y of the set {Sno point z ≠ y of the set {Snn | n = 1, 2, . . .}, | n = 1, 2, . . .},

belongs to V’. belongs to V’. This contradicts the fact that y is an accumulation This contradicts the fact that y is an accumulation

point of the set {Spoint of the set {Snn | n = 1, 2, . . .}, | n = 1, 2, . . .},

thus s=y.thus s=y.

EndnotesEndnotes1.1. Greever, John. Greever, John. Theory and Examples of Point-Set Topology.Theory and Examples of Point-Set Topology. Claremont: Claremont:

Waybread Publications, 1990. Waybread Publications, 1990. 2.2. Steen, Lynn Arthur and J. Arthur Seebach, Jr. Steen, Lynn Arthur and J. Arthur Seebach, Jr. Counterexamples in Counterexamples in

Topology. Topology. New York: Dover Publications Inc., 1995. New York: Dover Publications Inc., 1995. 3.3. Image of axiom spaces: http://jtauber.com/2005/01/separation.pngImage of axiom spaces: http://jtauber.com/2005/01/separation.png4.4. Sneddon, I. N. Ed. Andrew H. Wallace. “An Introduction To Algebraic Sneddon, I. N. Ed. Andrew H. Wallace. “An Introduction To Algebraic

Topology.” Topology.” International Series of Monographs in Pure and Applied International Series of Monographs in Pure and Applied Mathematics.Mathematics. V.I. New York: Pergamon Press, 1957. V.I. New York: Pergamon Press, 1957.

5.5. Baum, John D. Baum, John D. Elements of Point Set TopologyElements of Point Set Topology. New York: Dover . New York: Dover Publications Inc., 1991 Publications Inc., 1991

6.6. Sneddon, pg. 32.Sneddon, pg. 32.7.7. Goodman, Sue E. Goodman, Sue E. Beginning Topology. Beginning Topology. Belmont: Brooks/Cole, 2005. Belmont: Brooks/Cole, 2005. 8.8. Baum, pg. 81. Baum, pg. 81. 9.9. Image-http://en.wikipedia.org/wiki/Regular_spaceImage-http://en.wikipedia.org/wiki/Regular_space10.10. Image-http://commons.wikimedia.org/wiki/Image:Normal_space.svgImage-http://commons.wikimedia.org/wiki/Image:Normal_space.svg11.11. Steen, pg. 13.Steen, pg. 13.12.12. Steen, pg. 13.Steen, pg. 13.13.13. Sneddon, pg. 31.Sneddon, pg. 31.